The contractive product along one mode can be generalized to two modes resulting in an \((m-2)\) th order tensor, and it can preserve or compress (other than expand as a sequence of outer product) the tensors and is useful in many aspects. For example, the homogeneous polynomial \(f(\textbf {x}):=\mathcal {A}\textbf {x}^m\)  defined by (REF ) can be regarded as the contractive product of \(\mathcal {A}\)  with the rank-one tensor \(\textbf {x}^{m}\) , and \(\mathcal {A}\textbf {x}^{m-1}\)  is employed to define various eigen-pairs (see e.g. [1]}).
